One Real Root Conditions

Consider the cubic equation $x^3+ax+b=0$, where $a,b$ are real. Determine, with proof, which of the following conditions guarantee that the equation has exactly one real root.

1. $a=-3$, $b=-3$. 2. $a=-3$, $b=2$. 3. $a=-3$, $b>2$. 4. $a=0$, $b=2$. 5. $a=1$, $b=2$.